Compositional convection in the presence of a magnetic field. II

Proc. R. Soc. A (2005) 461, 2605–2633 doi:10.1098/rspa.2005.1473 Published online 28 June 2005

Compositional convection in the presence of a magnetic field. II. Cartesian plume B Y I. A. E LTAYEB 1 , E. A. H AMZA 1 , J. A. J ERVASE 2 , E. V. K RISHNAN 1 AND D. E. L OPER 3 1

Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, PO Box 36, Muscat 123, Sultanate of Oman ([email protected]) 2 Department of Electrical and Computer Engineering, College of Engineering, Sultan Qaboos University, PO Box 33, Muscat 123, Sultanate of Oman 3 Geophysical Fluid Dynamics Institute, The Florida State University, Tallahassee, FL 32306, USA The analysis of part I, dealing with the morphological instability of a single interface in a fluid of infinite extent, is extended to the case of a Cartesian plume of compositionally buoyant fluid, of thickness 2x 0, enclosed between two vertical interfaces. The problem depends on six dimensionless parameters: the Prandtl number, s; the magnetic Prandtl number, sm; the Chandrasekhar number, Qc; the Reynolds number, Re; the ratio, Bv, of vertical to horizontal components of the ambient magnetic field and the dimensionless plume thickness. Attention is focused on the preferred mode of instability, which occurs in the limit Re/1 for all values of the parameters. This mode can be either sinuous or varicose with the wavenumber vector either vertical or oblique, comprising four types. The regions of preference of these four modes are represented in regime diagrams in the (x 0, s) plane for different values of sm, Qc, Bv. These regions are strongly dependent on the field inclination and field strength and, to a lesser extent, on magnetic diffusion. The overall maximum growth rate for any prescribed set of the parameters sm, Qc, Bv, occurs when 1.3!x 0!1.7, and is sinuous for small s and varicose for large s. The magnetic field can enhance instability for a certain range of thickness of the plume. The enhancement of instability is due to the interaction of the field with viscous diffusion resulting in a reverse role for viscosity. The dependence of the helicity and a-effect on the parameters is also discussed. Keywords: compositional convection; compositional plumes; hydromagnetic stability; geodynamo; helicity; a-effect

1. Introduction In part I (Eltayeb et al. 2004), we considered the morphological instability of a vertical interface within an electrically conducting fluid of infinite extent. The fluid is composed of two components of differing density and the interface divides portions having differing amounts of the two components. Both portions have the same stabilizing thermal gradient and are threaded by a uniform applied Received 13 September 2004 Accepted 28 February 2005

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q 2005 The Royal Society

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I. A. Eltayeb and others

magnetic field. In the present study, we investigate the effect of a second vertical interface parallel to the first so that a channel of finite thickness, 2x 0, is formed. To facilitate discussions, the channel is assumed to contain fluid that is compositionally more buoyant than the surrounding fluid, although the analysis is valid for the opposite case. We shall refer to this model as the Cartesian plume. This study is motivated by a desire to understand better the morphological instability of plumes of compositionally buoyant material which may occur within Earth’s outer core (Loper 1983, 1987; Moffatt 1989). The prototypical shape of such plumes is cylindrical and various aspects of plumes of this shape have been the subject of a number of experimental (Copley et al. 1970; Sample & Hellawell 1984; Bergman et al. 1997; Classen et al. 1999; Jellinek et al. 1999) and analytic (Loper 1987; Eltayeb & Loper 1997; Worster 1997; Chung & Chen 2000; Morse 2000) studies. Cylindrical plumes are difficult to analyse, particularly when subject to rotation and hydromagnetic effects, and studies of simpler configurations, which are more amenable to analytic solution, serve to provide insights into the nature of their morphological instabilities. In particular, Eltayeb & Loper (1994, 1997) have shown that the morphological instabilities of Cartesian and cylindrical plumes, in the absence of rotation and hydromagnetic effects, are qualitatively similar. The effect of rotation on these instabilities has been studied by Eltayeb & Hamza (1998), while the effect of hydromagnetic forces are investigated in part I and in this study. Just as part I is a generalization of Eltayeb & Loper (1991), this study is a generalization of Eltayeb & Loper (1994), with the inclusion of hydromagnetic effects. The Cartesian plume has been shown to possess analytical solutions in the absence of hydromagnetic effects (Eltayeb & Loper 1994), and we shall see below that it does so also in their presence. The analytical solution allows us to examine the influence of the finite width of the plume together with the Lorentz force and magnetic diffusion. The morphological instability of the single interface depends on six dimensionless parameters—the Prandtl number, s, the magnetic Prandtl number, sm, the Chandrasekhar number, Qc, the Reynolds number, Re, together with Bv and G, which quantify the direction of the applied magnetic field: n sZ ; k

n sm Z ; h

Qc Z

B02 L2 ; mr0 nh

Re Z

UL ; n

Bv Z

Bz ; By

GZ

Bx ; By

(1.1)

where the ambient magnetic field is written in component form B0 Z B0 ðBx ; By ; Bz Þ;

Bx2 C By2 C Bz2 Z 1:

(1.2)

U is a characteristic velocity, L is a characteristic length-scale, n is the kinematic viscosity, k is the thermal diffusivity, h is the magnetic diffusivity, m is the magnetic permeability, r0 is the mean density and B0 is the total amplitude of the ambient magnetic field. The x-axis is normal to the interface, z is upward and y is horizontal and parallel to the interface (see figure 1). The analysis of part I included the effect of a small field normal to the interface (Gs0), but in the following analysis, this component is assumed to be zero, as its presence poses serious analytic difficulties, which will be discussed elsewhere. We thus assume that G Z 0: Proc. R. Soc. A (2005)

(1.3)

Hydromagnetic compositional convection

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Figure 1. A perspective view of the geometry of the problem drawn with representative profiles of basic (vertical) flow, wðxÞ and temperature, TðxÞ. The undisturbed vertical interfaces are the discontinuous lines at x ZGx 0. The y-axis is horizontal and parallel to the undeformed interfaces. The magnetic field is parallel to the vertical interfaces (and out of the plane of the paper).

In the present study of the Cartesian plume, the dimensionless thickness of the channel, 2x 0, is another parameter of the problem. The magnetic field has a profound effect on the instability of the single interface, with the nature of the effect depending on the orientation of the field. A horizontal magnetic field parallel to the interface has a stabilizing effect in the sense that the growth rate is reduced in magnitude, although the order of magnitude remains unchanged and the instability is not completely suppressed for any value of the parameters of the problem. On the other hand, the addition of a vertical component of the ambient field can counteract the influence of the horizontal field. The study of the non-magnetic Cartesian plume (Eltayeb & Loper 1994) has shown that the finite width of the plume leads to an enhancement of the instability. The study of the magnetic single interface (part I) has shown that an inclined magnetic field can also enhance instability. Our purpose here is to examine the influence of the three factors of (i) finite thickness of plume, (ii) the Lorentz force and (iii) magnetic diffusion, all acting simultaneously. The analysis below will examine the influence of these factors on the preference of these modes in the space (x 0, s, sm, Qc, Bv) when the Reynolds number Re is small. In §2, we briefly formulate the problem and present the solution. In §3, we discuss the stability results. As in the non-magnetic case (Eltayeb & Loper 1994), the morphological instabilities may be categorized as sinuous or varicose, depending on whether the symmetry of the solution in x is odd (sinuous) or even (varicose), and where necessary the symmetries will be distinguished by the Proc. R. Soc. A (2005)

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parameter P, defined by ( PZ

1

for even ðvaricoseÞ solution;

K1

for odd ðsinuousÞ solution:

(1.4)

Also, the horizontal wavenumber m can be zero (with the convective rolls aligned with the vertical) or non-zero (with the rolls being oblique). Consequently, four types of modes can occur: varicose oblique (labelled Vo), varicose vertical (Vv), sinuous oblique (So) and sinuous vertical (Sv). These modes can further be characterized by the effect of the magnetic field; we shall find that, as in the case of a single interface, these can be non-magnetic modes modified by the field or new modes introduced by the presence of the field. We shall see that the finite width of the plume enhances the instability; as the thickness approaches zero, the growth rate approaches zero as the square power of the thickness of the plume but always remaining positive. Consequently, for fixed values of s, sm, Qc, Bv the growth rate of the preferred mode attains an overall maximum at a certain value of x 0. It is also found that the horizontal component of the field acting alone can enhance instability for some range of values of x 0 and Qc, in sharp contrast to the corresponding case of a single interface. The addition of a vertical component of field leads to further enhancement of instability in the sense that the growth rate is increased in value but remains of the same order of magnitude. It will be shown that while magnetic diffusion is always stabilizing, its influence on the sinuous and varicose modes depends on the other parameters. The destabilizing influences of magnetic field and finite thickness of plume can sometimes act in concert to enhance instability but the two factors can at other times oppose one another. In §4, we discuss the helicity and a-effect of the perturbations, and in §5, we include a few concluding remarks.

2. The basic equations and boundary conditions Generalizing Eltayeb & Loper (1994), we shall consider an incompressible fluid of infinite extent flowing with velocity u in the presence of a stabilizing thermal gradient, gZdT/dz, and a uniform magnetic field B. The density, r, depends on the temperature, T and the concentration of light component, C. The fluid has finite kinematic viscosity, n, thermal diffusivity, k and magnetic diffusivity, h; material diffusivity is assumed to be zero, so that the distribution of C can be specified. We shall assume that it has the top-hat profile ( for jxj% x0; C0 C C CZ (2.1) C0 for jxjO x0: are constant in the absence of deformation of the interfaces, where both C0 and C (see figure 1). Both the compositional distribution and the interfaces are material functions, moving with the fluid. The formulation is given in part I. Here, we state the main results noting that all variables used below are dimensionless. The ~ , and non-dimensionalization process used the amplitude of the concentration, C that of magnetic induction, B0, as units of concentration of light material Proc. R. Soc. A (2005)


and magnetic induction, respectively, and ~ gL2 nk 1=4 bC LZ ; UZ ; agg n

tc Z

U ; L

~ gL Pd Z rbC

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(2.2)

as units of length, velocity, time and pressure, respectively. The Boussinesq approximation, in which density variations are neglected except when they occur in the gravity term in the equation of motion, is adopted. As a consequence, gravity is the driving force for the instability and is represented by the terms C and T in equation (2.8). The positions of the two interfaces are assumed to be slightly disturbed, with locations described by xv ZGx0 G3 exp½Ut C iðmy K nzÞ C c:c:

(2.3)

for the varicose mode and by xs ZGx0 C 3 exp½Ut C iðmy K nzÞ C c:c:

(2.4)

for the sinuous (or meandering) mode, where c.c. is the complex conjugate and 0!3/1. This disturbance is accompanied by dimensionless convective perturbations of velocity, u*, pressure, p*, magnetic field, b*, temperature, T * and concentration of light material, C *, of the form ( ) Kinu;Knv; w;Kinp; fu ;p ; b ;T ;C g Z3 exp½Ut Ciðmy KnzÞ Cc:c: (2.5) Kinbx ;Knby ; b; T;C The perturbation equations are ~ Du C idQbx K Dp Z ReUu;

(2.6)

~ Dv C idQby C mp Z ReUv;

(2.7)

~ K inDwu; Dw C idQb C C C T C n 2 p Z Re½Uw

(2.8)

~ K inDTu; DT K w Z sRe½UT

(2.9)

~ x; Dbx C idu Z Resm Ub

(2.10)

~ m by ; Dby C idv Z ReUs

(2.11)

~ C inDwb x ; Db C idw Z Resm ½Ub

(2.12)

C Z 0;

(2.13)

Du C mv C w Z 0

(2.14)

Dbx C mby C b Z 0;

(2.15)

and

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in which D hd=dx;

D hD2 K m2 K n2 ;

d Z m K Bv n;

(2.16)

~ 1 Z U1 K inwðxÞ; U

(2.17)

Qc : 1 C Bv2

(2.18)

and QZ

The functions wðxÞ and TðxÞ represent the basic state vertical plume flow and temperature (see figure 1) and are given by 1 (2.19) wðxÞ Z ½expðKXCÞsinðXCÞ K expðKjXKjsinðXKÞ; 2 1 TðxÞ Z f½expðKXCÞcosðXCÞ K1K½expðKjXKjÞcosðXKÞ K1sgnðXKÞg; (2.20) 2 in which jxjGx XG Z pffiffiffi 0 : 2

(2.21)

As in part I, the following analysis is somewhat simplified by the introduction of Q and d in place of Qc and Bv. The parameter Q always occurs in the combination d2Q (or its square root) and such a combination is a local Chandrasekhar number based on the component of the magnetic field parallel to the perturbation wavenumber d2 Q Z

ðk$BÞ2 L2 ; mr0 nh

(2.22)

where kZV(myKnz). If the waves are aligned with the field, then k $BZ0, and ~ U, where U is the field has no influence on the instability. Note that ReUZ defined by eqn (4.9) in Eltayeb & Loper (1991), taking into account the difference in the sign of n between the two analyses. However, the results below will be presented using Qc because it is independent of the direction of field and the wavenumbers. The continuity equation may be replaced by an equation governing the variation in pressure, obtained by forming D(2.14) and making use of (2.6)–(2.8), (2.14) and (2.15): Dp K T Z 2iRenDwu:

(2.23)

Now (2.6), (2.8)–(2.10), (2.12) and (2.23) form a set of six equations for the six unknown variables u, w, p, T, bx and b. Equations (2.7) and (2.11), and the associated variables v and by are decoupled and not of interest. However, note that the three-dimensionality of the solution is retained due to the presence of the factor m in (2.16) and (2.17). The boundary conditions are the continuity of momentum, heat and magnetic field fluxes together with the condition that the interfaces are material surfaces, Proc. R. Soc. A (2005)


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Figure 2. Illustration of the two modes of the interfaces. The two vertical interfaces are here situated at x ZG0.5 and the disturbance has an amplitude of 0.2. The solid curves refer to the even mode (varicose wave) and the dash-dotted curve on the right with the continuous one on the left give the odd mode (sinuous or meandering wave) and the concentration of light material within the greater than outside. plume is C

and that all the perturbation variables decay to zero away from the interface. In addition, the continuity of v and w can be used in (2.14) to show the continuity of Du across the interfaces. Similarly the continuity of by and b implies that of Dbx according to equation (2.15). Furthermore, equation (2.12) can be integrated to find that Db across the interface making use of the continuity of u, b, w and Dw is also continuous across the interface. The relevant boundary conditions can then be written as: 9 ðiÞ u; w; p; T; bx ; by ; b/ 0 as x /GN; > > > > ðiiÞ u; w; p; T; bx ; b; Du; DT; Dbx ; Db are continuous at x ZGx0 ; = (2.24) ðiiiÞ DwðGx0KÞ K DwðGx0CÞ Z 1; DpðGx0KÞ K DpðGx0CÞ Z K1; > > > > ; Gx0 Þ: ðivÞ K inuðGx0 Þ Z U K in wð The full solution to the problem is a linear combination of two solutions having opposite symmetries in x (see figure 2). In the even (varicose) solution w, p, T, v, by, b are even in x and u, bx are odd. The reverse is true for the odd (sinuous) solution. We shall then restrict our analysis to the half-interval [0,N) and use the following parity conditions at xZ0: Dw v ð0Þ Z Dpv ð0Þ Z DT v ð0Þ Z Dbv ð0Þ Z u v ð0Þ Z bvx ð0Þ Z 0 Proc. R. Soc. A (2005)

(2.25)

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for the varicose solution and w s ð0Þ Z ps ð0Þ Z T s ð0Þ Z bs ð0Þ Z Du s ð0Þ Z Dbsx ð0Þ Z 0

(2.26)

for the sinuous solution. The variables in (2.6)–(2.15) are assumed to have the expansion f ðx; y; z; tÞ Z

N X

fr ðx; y; z; tÞRer ;

(2.27)

rZ0

UZ

N X

Ur ReðrK1Þ ;

(2.28)

rZ1

in terms of the small parameter Re. Now the problem may be solved iteratively starting with the zeroth-order solution obtained by setting ReZ0. The stability of the interface is determined by the sign of the real part of the dominant term in the expansion for U. If Re(Ur)!0 for all possible values of wavenumbers m and n, the interface is stable to harmonic perturbations. If, on the other hand, Re(Ur)O0 for any pair of wavenumbers (m, n), then the plume is unstable. The solution is obtained by substituting the expansions (2.27) and (2.28) into equations (2.6)–(2.15) and equating the coefficients of Re r , (rZ0, 1, 2, 3, .) to zero to obtain a hierarchy of systems of equations which can be solved seriatim. We will only need to consider the first two such problems in order to determine the growth rate to leading order. The zeroth-order set of equations is obtained by setting the right-hand side to zero in equations (2.6)–(2.12) together with equations (2.14) and (2.15). The boundary conditions are those in (2.24). The solution may be expressed as " ( l ðxKx Þ )# 3 0 ej 1X fw0 ; T0 ; p0 ; b0 g Z ; (2.29) Aj fm3j ; m2j ; mj ;Kidm2j g P eKlj ðxCx0 Þ C 2 jZ1 eKlj ðxKx0 Þ " ( l ðxKx Þ )# 3 0 Ke j fm2j ;Kidmj g 1X Klj ðxCx0 Þ fu 0 ; bx0 g ZK ; Aj lj 2 C P e 2 jZ1 mj Cd2 Q eKlj ðxKx0 Þ

(2.30)

where Aj Z

m2j Cd2 Q ; 2lj ½2mj ð1 Cd2 QÞ C3n 2

m2j Cð1 Cd2 QÞmj Cn 2 Z 0; lj Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mj Cm 2 Cn2 :

(2.31)

(2.32)

The upper (lower) expression in the curly brackets refers, respectively, to the subinterval 0%x!x 0(xOx 0) and P is defined by equation (1.4). The leading Proc. R. Soc. A (2005)


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order contribution to the growth rate U1 is non-zero, 0Þ C U1 ðm; n; x0 ; PÞ Z inwðx

3 Aj lj m2j inP X expðK2lj x0 Þ 2 jZ1 m2j Cd2 Q

(2.33)

and the unstable waves must propagate. The phase speeds in the vertical and lateral directions are given, respectively, by U1 U1 U n nU Uz Z ; Uy ZK ZK 1 ZK z : (2.34) in im in m m The first-order equations (i.e. those terms having coefficients of Re1) are nonhomogeneous, with the homogeneous part being the same as the zeroth-order problem. The solution must obey a solvability condition that provides an expression for the second term, U2, in the growth rate. The details of the derivation of the expression for U2 are given in appendix A. It can be written in the form U2 Z U20 CsUsi Csm Usm ;

(2.35)

where U20, Usi, Usm are given in (A 26)–(A 28). The three terms on the right-hand side of equation (2.35) correspond to the contributions of thermal, viscous and magnetic diffusions. The expressions (2.34) for the phase speeds and (2.35) for the growth rate will be discussed in §3 below. In that discussion we will revert to consideration of the externally imposed parameters Qc and Bv, making use of equations (2.17) and (2.18). 3. The instability of the magnetic Cartesian plume The phase speeds (2.34) and (2.33), which are independent of s and sm, have been computed as functions of the parameters Qc, x 0 and Bv and the wavenumbers m and n. It was found that the lateral phase speed is very small everywhere except near mZ0, and is negative for the varicose mode. The contours of the phase speed for the sinuous mode show that Uy is negative except in some small patches of small n where it is positive. For the varicose mode, when BvZ0, Uz is positive and its largest value occurs when mZ0. As Qc increases, Uz increases slowly, but an increase in x 0 leads to appreciable reduction for ms0. Except for small values of x 0, Uz for the varicose and sinuous modes have 0 Þ is small for x 0R1.0. opposite signs as wðx The growth rate U2, given by equation (2.35), is a complicated function of the parameters m, n, x 0, s, sm, Qc, Bv. Our purpose here is to identify the maximum growth rate (maximized over m and n) and delineate those regions in the five parameter space (x 0, s, sm, Qc, Bv) where the maximum is positive, leading to instability, and negative signifying stability. As there are two categories of modes (determined by PZ1 for varicose and PZK1 for sinuous modes) we need to determine the maximum for each symmetry and then choose the larger of the two. In doing so, we shall denote the growth rate (2.35) by UP2 . The growth rate is maximized over the wavenumbers m and n by setting vUP2 =vmZ vUP2 =vnZ 0. The maximum for each mode will be denoted by UPm and the associated Proc. R. Soc. A (2005)

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Figure 3. A sample of the results for UPm exhibiting the main general features of the stability problem. The solid curves refer to the varicose mode while the dashed ones refer to the sinuous mode. (a) QcZ1.0, sZsmZ0.0 and BvZ0, 0.5 as labelled; (b) QcZBvZ1.0, smZ0.0 and sZ0.0, 5.0 as labelled; (c) x 0Z2.0 and (Bv, s, sm) as labelled; (d ) x 0Z3.5 and (Bv, s, sm) as labelled.

wavenumber components are mm and nm and we shall refer to the associated parameter dm, which measures the inclination of the wave vector to the field lines. The larger of the two growth rates UPm is denoted by Uc and the preferred (critical) mode of convection is defined by the maximum growth rate and the associated wavenumbers together with d and is referred to as (Uc, mc, nc, dc). The results obtained for the single interface in part I provide a check on these results for x 0/N and those for QcZ0 can be compared with those of the non-magnetic Cartesian plume (Eltayeb & Loper 1994). In view of the complexity of the problem in its entirety we will represent the most significant results graphically. It is found that the influence of the parameters is very dependent on the symmetry (varicose or sinuous). The effect of the different parameters can be conflicting for some values and can act in concert at other values. In figure 3, we present a sample of the results for UPm for some representative values of the parameters. In figure 3a,b, the dependence of the growth rate for the two symmetries of modes as functions of the thickness of the plume, x 0, is illustrated for sample values of s, sm, Qc, Bv. In all cases, the growth rate acquires a maximum at a finite value of the thickness of the plume and approaches the value of the single line interface when x 0 is very large. For the sinuous mode, the growth rate increases from 0 at x 0Z0.0 until it reaches Proc. R. Soc. A (2005)


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a maximum before it decreases steadily to its asymptotic value at large values of x 0. In the case of the varicose mode, the growth rate increases from 0 at x 0Z0.0 to a maximum and then decreases to a finite positive minimum before it increases slowly to its asymptotic value for large x 0. In the absence of viscous and magnetic diffusions, as in figure 3a, the sinuous mode is preferred except for small plume thickness and the influence of increasing the field inclination, as represented by Bv, is to reduce the growth rate for both modes. In figure 3b, the influence of increasing viscous diffusion is illustrated. It is evident that the effect of viscous diffusion on the two modes depends on the thickness of the plume. Viscous diffusion can be stabilizing or destabilizing depending on the mode and on the thickness of the plume. This interesting feature will be examined in more detail later. In figure 3c,d, we illustrate the growth rate as a function of Qc for different sets of the parameters x 0, s, sm, Bv. It is evident that the preference of either mode is a complicated function of the parameters. In all cases we see that magnetic diffusion, as represented by sm, is always stabilizing for both categories of modes. In figure 3c, x 0Z2.0 and the influence of s is stabilizing for the sinuous mode and destabilizing for the varicose mode. This leads to the preference of the sinuous mode for small s and the varicose mode for large s (see figures 4–8). The influence of the field inclination is more complicated. For every non-zero Bv, the growth rate of the sinuous mode decreases with Qc and the rate of decrease increases with the increase in Bv. For the varicose mode, the growth rate can increase or decrease with the increase in Qc depending on Bv. In figure 3d, x 0Z3.5, the effect of viscous diffusion is always stabilizing. The growth rates of both categories of modes decrease with Qc for all Bv, but the rate of decrease depends on the value of the inclination parameter. The sinuous mode is always preferred here. We will discuss these properties in more detail below. As noted previously, the preferred mode is one of four modes: vertical sinuous, Sv, oblique sinuous, So, and the corresponding varicose modes, Vv and Vo. The influence of the different parameters on these four modes results in the preference of one mode for particular ranges of parameters. In view of the number of parameters and number of modes, we find it informative to represent the preference of modes in a regime diagram in the x 0Ks plane. This is illustrated in figures 4–8. In addition to the identification of the regions of preference of the different modes, we have also included the curve of the overall maximum growth rate, Umax, obtained by maximizing the critical growth rate as a function of s and x 0 , i.e. by locating the wavenumbers and growth rate satisfying vUc =vx0 Z vUc =vsZ 0. The curve of overall maximum growth rate is represented by a discontinuous curve in the regime diagrams. We note that it does not change very much with changes in s, sm, Qc, Bv. In figure 4, the influence of increasing the amplitude of a horizontal field in the absence of magnetic diffusion is illustrated. Starting with the non-magnetic case of QcZ0.0, Qc is increased gradually and the associated evolution of the regime diagram in the x 0Ks plane is monitored. The influence of increasing Qc here is to suppress the vertical varicose mode for small x 0 and moderate s. The vertical sinuous mode is suppressed for small x 0 and moderate s and extended for large x 0 and moderate s. The region of the oblique varicose mode remains almost unaffected by the increase in Qc. Proc. R. Soc. A (2005)

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Figure 4. The regime diagram in the x 0Ks plane illustrating the regions where the different modes are preferred for a horizontal magnetic field (BvZ0.0) when smZ0.0. (a – d ) correspond to Qc taking the values 0.0, 0.20, 1.0 and 100.0, respectively. The dashed curve refers to the position of the global maximum with growth rate Umax. We note that the horizontal magnetic field tends to suppress the vertical modes for small values (%1.2) of x 0 and enhance them for large x 0. The vertical wavenumber is about unity for the vertical mode when x 0 is small and is less than unity for all other modes. The horizontal wavenumber, on the other hand, is rarely in excess of 0.4.

Some of the most interesting features of the stability of the problem are those brought about by the presence of magnetic diffusion and the vertical component of field. Figure 5 illustrates the evolution of the regime diagram with the increase of the field-inclination parameter, Bv, when magnetic diffusion is absent. It would have been expected that the presence of the vertical field will suppress the vertical modes as it will tend to align the waves with the inclined magnetic lines of force. This is found to be the case for the varicose mode, although even in this case only the vertical mode for small x 0 is affected while the oblique varicose mode present for moderate x 0 is unaffected. However, the effect on the sinuous mode is different. For small field inclination, the region in the x 0Ks plane where the vertical sinuous mode is preferred begins to shrink so that when Bv increases to about 0.02, Sv is present only in two small pockets: one for small x 0 and another for large x 0 (see figure 5b). As Bv increases further, the pocket for large x 0 shrinks further until it eventually disappears, while Proc. R. Soc. A (2005)


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Figure 5. The x 0Ks regime diagram in the case QcZ1.0, smZ0.0 illustrating the effect of increasing the vertical component of field as follows: (a) BvZ0.02, (b) BvZ0.1, (c) BvZ0.2, (d ) BvZ0.5. Compare (a) with figure 4c for BvZ0.0. While the varicose vertical mode disappears once the vertical field is non-zero, the regions of preference of the sinuous mode are only partly suppressed for small vertical fields as in (a). Further increase in Bv leads to further suppression of the sinuous vertical mode for large x 0 but at the same time a new sinuous vertical mode appears for smaller (less than about 1.2) x 0 and sR4.0. This new vertical mode region expands into a smaller x 0 direction and eventually extends to the region for oblique varicose mode present for small x 0 (see panel (c)). Further increase in Bv to approximately 0.4 leads to the migration of this region towards the s-axis, and to the complete disappearance of the vertical mode. Note that the dashdotted curves represent a sudden change in the horizontal wavenumber without change of mode and they normally signify the merging of two different modes for smaller Bv, while the dashed ones refer to the position of the maximum growth rate.

the pocket for small x 0 extends and moves towards the s-axis and remains for all values of the inclination Bv. This latter vertical sinuous mode is magnetic in nature brought about by the presence of the field. When magnetic diffusion is present, the regime diagram evolves in the same way as regards the varicose modes but the sinuous mode evolves differently in the sense that the vertical sinuous mode is pushed further into regions of large s for both small and large x 0 (see figure 6). The influence of magnetic diffusion in the absence of the vertical field is illustrated in figure 7. An increase in magnetic diffusion leads to a change in all Proc. R. Soc. A (2005)

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Figure 6. The influence of the vertical field component on the regime diagram in the presence of magnetic diffusion. Here QcZ1.0, smZ1.0 and (a) BvZ0.0, (b) BvZ0.02, (c) BvZ0.10, (d ) BvZ0.50. Compare with figure 4 and note the difference in the regions of preference of Sv when x 0 exceeds about 3.0.

of the regions (in the x 0Ks plane) of preference of the four modes Sv, So, Vv, Vo. In general, an increase in magnetic diffusion leads to an increased preference for vertical modes. The vertical sinuous mode region expands into areas of large s. The region of preference of the sinuous oblique mode is pushed towards the s-axis until it eventually disappears at some value of sm when Qc is small but will persist as a thin region along the s-axis for large Qc. The increase in sm from 0 also leads to a new region of preference of a vertical varicose mode which appears as part of the left side of the region of preference of the oblique varicose mode. This new region of Vv eats away the oblique varicose mode as sm increases (see figure 7). In the presence of the vertical field the influence of magnetic diffusion is, in general, dependent on Qc (see figure 8). For small Qc, the vertical sinuous mode present in the absence of diffusion is suppressed by the presence of magnetic diffusion. If, however, Qc is large and the vertical sinuous mode is absent in the absence of diffusion, an increase of magnetic diffusion will lead to the appearance of a thin region of Sv adjacent to the s-axis. While the oblique varicose mode Proc. R. Soc. A (2005)


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Figure 7. The influence of magnetic diffusion on the x 0Ks regime diagram in the presence of a horizontal magnetic field. Here QcZ10.0, BvZ0.0 and (a), (b) and (c) refer to smZ0.0, 5.0 and 10.0, respectively. Note that the dashed curve refers to the position of the maximum growth rate.

present for small x 0 is suppressed near the s-axis, the oblique varicose mode for moderate x 0 is unaffected by an increase in sm. The most prominent change in the regime diagram is the appearance of a vertical sinuous mode in a thin region near the s-axis when sm increases beyond a certain value dependent on Qc (see figure 8). The influence of magnetic diffusion and the presence of the vertical field on the growth rate of the preferred mode lead to some novel features of instability. Figures 9 and 10 depict the profiles of Uc as a function of the vertical field inclination Bv for different values of Qc. In figure 9, the growth rate of the preferred mode of the oblique varicose mode is shown as a function of Bv for sZ5.0, x 0Z1.8 and different values of sm. For small values of sm, as Bv increases from 0, the growth rate decreases gradually until it reaches a minimum, Uc0, at a value, Bv0 of Bv before it increases again gradually to a maximum and then starts to decrease steadily with the increase of Bv (see figure 9a,b). The value Uc0 is the same as that obtained in the absence of the field and the critical mode there is identical to that of the non-magnetic mode so that dcZ0. For moderate and large Qc, the growth rate increases as Bv increases from 0 until it reaches a maximum, Ucm, at BvZBm1, before it decreases to a minimum, Uc0, at BvZBv0 after which it increases again to a maximum, Ucm, at BvZBm2 and then decreases steadily with Bv. The local maximum values of the preferred growth rate at BvZBm1, Bm2 are always identical whatever the value of Qc, and Uc0 is the same or greater than Proc. R. Soc. A (2005)

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Figure 8. The influence of magnetic diffusion on the x 0Ks regime diagram when the field is inclined to the vertical. Here, BvZ0.5 and (a) QcZ1.0, smZ0.0, (b) QcZ1.0, smZ5.0, (c) QcZ10.0, smZ0.0 and (d ) QcZ10.0, smZ5.0. The increase in magnetic diffusion here suppresses the vertical sinuous mode when Qc is small and gives rise to it for small x 0 when Qc is large. The dashed curve again refers to the position of the overall maximum.

the corresponding value for QcZ0. As Bv increases from 0, dc decreases steadily through positive values until Bv reaches Bv0. Here, dcZ0 if Qc is less than some value Q0(s,sm) dependent on s and sm, in general. For QcOQ0, dc jumps to a negative value and further increase in Bv leads to a steady decrease in dc, with the rate of decrease diminishing with increasing Qc. If, on the other hand, sm is moderate or large, the preferred growth rate increases with field inclination reaching a maximum Uc0 identical with that in the absence of the field before it starts to decrease steadily with Bv. The values of Ucm and Uc0 depend on Qc, s and sm. The remarkable result is that for small sm, Ucm exceeds the growth rate of the corresponding nonmagnetic case and consequently the vertical field destabilizes the plume. Furthermore, if the magnetic field strength, as measured by Qc, is in excess of a certain value Q0(s,sm), both Ucm and Uc0 are greater than the growth rate in the absence of the field. However, an increase in magnetic diffusion leads to the suppression of Ucm, but Uc0 remains for all non-zero values of Qc and occurs for a value of Bv that depends weakly on sm. The wavenumbers and dc associated with these profiles are shown in figure 10. We note that the change of mode at Bv0 is indeed associated with a drop in both vertical and horizontal wavenumbers, Proc. R. Soc. A (2005)


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Figure 9. An illustration of the conflicting influences of the vertical field and magnetic diffusion on the preferred mode of convection as experienced at a point (1.8, 5.0) in the oblique varicose mode region in the regime diagram. Here (a) smZ0.0, (b) smZ0.5, (c) smZ1.0 and (d ) smZ4.0, and the Arabic numerals refer to the value of Qc. We note that in the absence of magnetic diffusion (i.e. when smZ0) the vertical field component can enhance instability (as in panel (a)), but an increase in magnetic diffusion acts to suppress this destabilizing influence so that when sm is about unity, the enhancement of growth rate by the vertical component of field has completely disappeared and thereafter the maximum possible growth rate for field strengths is the value in the absence of the field. Such a value can only occur for a certain value of Bv, which depends on s. The wavenumbers experience a jump for the modes with the two local maxima. However, for the cases which touch the line for QcZ0, the wavenumbers vary continuously (see figure 11).

but this occurs in such a way that dc merely changes sign with the numerical value remaining the same. As Bv increases from 0, dc decreases steadily towards 0. For values of Qc where there is no change of mode, dc decreases continuously into negative values. However, the cases in which there is a change of mode are associated with a jump in dc from positive to negative values (figure 11). The simultaneous action of the vertical field and magnetic diffusion on the preferred mode depends on the category mode. This is illustrated in figure 10 for both varicose and sinuous modes in the regions where they are preferred. For values of sm less than about half s, the varicose mode decreases from its value at BvZ0 to a minimum before it increases to a local maximum and then decreases steadily thereafter. For larger values of sm, the growth rate increases to a maximum before it starts to decrease steadily with Bv (see figure 10a,c). The sinuous mode behaves differently. The influence of the vertical field is potent only Proc. R. Soc. A (2005)

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Figure 10. Comparison of the effect of magnetic diffusion on the varicose and sinuous modes in the regions where they are preferred shown here for two values of Qc when sZ10.0; and (a – d ) shown for (Qc, x 0, P) corresponding to (2.0, 1.8, 1), (2.0, 3.5, K1), (10.0, 1.8, 1) and (10.0, 3.5, K1), respectively. The Arabic numerals refer to the values of sm. We note that magnetic diffusion has a strong stabilizing influence on both types of mode. However, the influence on the two modes is different in the sense that the preferred sinuous mode is influenced only when the field’s inclination to the vertical is small. The ‘hump’ curves in (b) and (d ) correspond to magnetic modes in which jdj is relatively large.

for small inclinations of the field and even then it does not appear to affect the vertical wavenumber if magnetic diffusion is not too large (i.e. sm is less than about half s). As the magnetic field inclination is increased from 0, the critical growth rate will decrease slightly if smZ0 before a change of mode takes place, in which case the horizontal wavenumber is reduced to 0 while the vertical wavenumber remains almost unchanged (see figure 12). The new mode is associated with a growth rate that increases with inclination reaching a maximum before it gradually decreases to a minimum where another change of mode takes place. This new mode is associated with a relatively large value of dc (see figure 12d ) and is magnetic in nature. Further increase in Bv leads to an increase in the growth rate to an overall maximum, Ucc, and thereafter decreases steadily. For moderate non-zero values of sm, similar behaviour occurs for the sinuous mode. The values of Bv between the two changes of mode are associated with a relatively large value of dc. As sm exceeds about half s, the growth rate increases steadily with inclination reaching the same overall maximum occurring for smZ0 and thereafter decreases steadily with inclination. For values of sm Proc. R. Soc. A (2005)


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Figure 11. The critical mode parameters (Uc, mc, nc, dc) as a function of the vertical field inclination Bv for x 0Z1.8, sZ5.0, smZ0.0 for different values of Qc (as labelled). This is a case in which the vertical field enhances the growth rate in a certain range of inclination. We observe the changing profile of the growth rate as Qc increases.

close to s, the growth rate again experiences a change of mode close to BvZ0 before it increases steadily reaching and following the values for moderate sm when Bv reaches values beyond about 0.2. This last change of mode is associated with a large drop in wavenumbers, as well as in dc (see figure 12). It then follows that the growth rate of the critical mode of the varicose mode is affected by the presence of magnetic diffusion at all inclinations of the magnetic field while that of the sinuous mode is affected only when the inclination is small. In an attempt to understand the mechanism leading to the enhancement of the growth rate by the magnetic field, we investigated the dependence of the individual contributions UP20 , UPsi , UPsm (see equation (2.35)) to the growth rate in detail. The destabilizing influence of the magnetic field can be traced down to the component UPsi of the growth rate due to viscous diffusion. In figure 13, we plot the maximum UPsi;m of UPsi and the associated wavenumbers mm, nm as functions of Qc for the case x 0Z1.8 in the absence of the vertical field. We find that UPsi;m initially increases with Qc for both even and odd modes and that it is positive in both cases, but its values for the even mode are about seven times those for the odd mode. Although the thermal diffusion contribution UP0;m has a maximum with mmZ0.0 and hence is aligned with the magnetic field, the maximum of the viscous contribution is inclined to the field. The magnetic diffusion contribution UPsm is negative for all m and n and has maximum 0 with mmZ0.0. We then Proc. R. Soc. A (2005)

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Figure 12. The critical mode parameters (Uc, mc, nc, dc) as a function of the vertical field inclination Bv for QcZ2.0, x 0Z3.5, sZ10.0, for different values of sm as labelled. The critical mode here is sinuous.

conclude that the destabilization of the plume by the horizontal field is a result of the magnetic field tending to inhibit the stabilizing influence of viscosity. In the presence of the vertical component of field, a similar detailed examination of the contributions UP0 , UPsi , UPsm to the growth rate reveals that the viscous contribution UPsi interacting with the vertical component is responsible for the enhancement of the growth rate. This is illustrated in figure 14. We see that UPsi increases with Bv as it increases from 0 for both modes. However, U1si reaches a maximum and then decreases steadily, but slowly, with Bv, while UK1 si increases slowly and then more rapidly for a short interval before it continues to increase slowly again. This change of pace with Bv in the case of the odd (sinuous) mode is due to a change of mode (see figure 14b,c). For both modes the inclined field tends to suppress the stabilizing influence of viscosity. The two local maxima at Bm1 and Bm2 (see figure 9) are associated with values of d of equal numerical value but different signs, because Uc depends on the magnetic field through the presence of d2. This means that they are associated with positions equally placed on either side of the direction dZ0. The result that the position dZ0 is not preferred, and indeed the system avoids it as it jumps from one side of dZ0 to the other, indicates that the interaction of viscous diffusion and the vertical component of the magnetic field can act in concert to destabilize the system through a wave that propagates at an angle to the magnetic field. The resulting growth rate is larger than that of the nonmagnetic system when the basic parameters are the same. Proc. R. Soc. A (2005)


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Figure 13. The maximum value of the viscous contribution (a) UPsi;m and (b) the associated wavenumbers, when the magnetic field is horizontal and x 0Z1.8 for the even (i) and odd (ii) modes. Note that the growth rates for both modes initially increase with Qc; the even mode acquiring a maximum at QcZ5.0 with mZ0.57 and nZ0.391, while the odd mode increases beyond QcZ10.0. The vertical wavenumber decreases for the even and increases for the odd modes, while the vertical wavenumber decreases for almost all Qc for both modes.

The perturbation equations (2.6)–(2.15) can be used to obtain an expression for the growth rate in terms of energy integrals (see Eltayeb & Loper 1991; eqns (4.40)–(4.42)). Omitting the details, we find that the growth rate depends on the three integrals (in addition to others) ðN ðN ðN c c c T dx; (3.1) x b dx; I3 Z s ðKDTÞu I1 Z Dwuw dx; I2 Z sm Dwb 0

0

0

representing the contributions of the basic state gradients to the growth rate. Here the superscript c refers to the complex conjugate. Computations of these integrals showed that I1, representing thermal diffusion, is positive while I2, representing magnetic diffusion, is negative signifying the stabilizing role of magnetic diffusion. The integral I3 representing the viscous diffusion contribution resulting from the interaction of the wave with the lateral variations of temperature can take positive and negative values. The enhancement of the growth rate by the magnetic field corresponds mainly to the positive values of this integral. It then follows that the instability is enhanced by the release of energy by the basic state temperature variations resulting from the rapid change in the composition of the light material across the interface. Proc. R. Soc. A (2005)

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Figure 14. The growth rate contributions UP0 , UPsi when x 0Z1.8, QcZ5.0. The curves i, ii, iii and iv 1 K1 refer, respectively, to U10 , UK1 0 , Usi , Usi in (a), and their associated mm, nm, dm shown in (b – d ), respectively. We note the similarity of (b) and (d ) with figure 12b,d.

4. Helicity and a-effect The expressions for the helicity and a-effect (see equation (4.2) of part I) can be written in the form 1 H ZnRe nðu v Cuv Þ Kmðu w Cuw Þ K ½ðw Dv CwDv ÞKðv Dw CvDw Þ ; 2 (4.1) n Ey ZK Imfðub Ku bÞCðwbx Kw bx Þg; 2

(4.2)

n Ez ZK Imfðuby Ku bÞ Cðvbx Kv bx Þg; 2

(4.3)

Ey CBv Ez E Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 CBv2

(4.4)

Here Ey, Ez represent the components of the a-effect along the horizontal and vertical components of the ambient field while E represents the a-effect along the inclined field. The leading order a-effect and helicity are obtained by Proc. R. Soc. A (2005)


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Figure 15. The profile of the a-effect and its components for a set of the parameters s, sm, Qc, Bv, m, n for varicose (a), and sinuous (b), modes. The roman numerals i, ii, iii, iv, v correspond, also in figure 16, to the set values, (1.0, 0.1, 5.0, 0.0, 0.139, 0.427 2), (1.0, 0.5, 5.0, 0.0, 0.137, 0.431 5), (1.0, 0.5, 5.0, 4.0, 0.206, 0.454 3), (1.0, 0.5, 2.0, 0.0, 0.24, 0.511 5), (10.0, 0.5, 5.0, 0.0, 0.192, 0.440 5), respectively, for the varicose mode, as in (a – c). For the sinuous mode, as in (d – f ), the corresponding values for s, sm, Qc, Bv are the same but (m, n) take the corresponding values (0.031, 0.609 5), (0.23, 0.553), (0.225, 0.533 5), (0.227, 0.589 1), (0.258, 0.529 2), respectively. Note that i, ii and iii correspond to a preferred varicose mode while iv an v correspond to a preferred sinuous mode. The dotted vertical line is the undisturbed interface.

using equations (2.29) and (2.30). They are illustrated graphically in figures 15 and 16. It is clear from equations (4.1)–(4.4) that the helicity and a-effect are both odd in x and we can therefore illustrate them in the half-interval [0,N). The expressions (4.1)–(4.4) also show that the helicity is discontinuous across the interface, since Dw0 is discontinuous there, and both components of the a-effect are continuous across the interface. The profiles of the components of the a-effect show clear dependence on the parameters s, sm, Qc, Bv and they are associated with variations on the small scale of the plume. They are illustrated in figure 15. The amplitude of the sinuous mode is about half that of the varicose mode for both components. However, there are marked differences between the dependence of the varicose and sinuous Proc. R. Soc. A (2005)

2628


Figure 16. The profiles of the helicity for the cases i and iv of figure 15. The other cases are not shown because they do not show any significant variation. For the varicose mode (a), all of ii, iii and v lie between the curves shown while in the case of the sinuous mode (b), they almost coincide with curve iv.

modes on the parameters s, sm, Qc, Bv. For the varicose mode, both components show strong dependence on variations in Bv. An increase in Qc leads to a reduction in the amplitude, while changes in s or sm have little influence on the amplitude. For the sinuous mode, on the other hand, changes in Qc or s are associated with notable changes in amplitude, while changes in Bv and sm have little influence on the amplitude. The profile of the helicity depends on the parity. For the varicose mode, the helicity decreases steadily from 0 at the centre of the plume until it reaches the interface where it suffers a positive jump. For xOx 0, the helicity for the varicose mode decreases rapidly to 0. Variations in the parameters s, sm, Qc, Bv have little effect on the profiles of the varicose mode. The most notable variations, albeit very small, occur near the interface within the plume. Here, an increase in s reduces the amplitude while an increase in sm or Bv tends to increase the amplitude. In the case of the sinuous mode, the helicity increases from 0 at the centre of the plume reaching a maximum before it decreases to a negative value on the interface. At the interface, it experiences a positive jump and thereafter it decreases rapidly to 0. The profile of the helicity of the sinuous mode is weakly dependent on s, sm, Qc, but shows clear dependence on variations in Bv; both the local maximum within the plume and the jump across the interface increase with Bv (see figure 16). 5. Concluding remarks The morphological instability of a planar plume of buoyant fluid, having thickness 2x 0 in an infinite medium permeated by a magnetic field, has been studied. The maximum growth rate of the instability is identified for the space of parameters x 0, s, sm, Qc, Bv, and the results have been compared with the two Proc. R. Soc. A (2005)


2629

related limiting cases of a single interface in the presence of a magnetic field (part 1) and the Cartesian plume in the absence of a field (Eltayeb & Loper 1994). These comparisons have revealed that the magnetic field and the finite thickness of the plume both have a profound effect on the stability problem. Magnetic diffusion, quantified by sm, is always stabilizing, but the strength of the stabilizing influence depends on the symmetry or parity of the mode of instability, which may be either sinuous (odd) or varicose (even). An increase in the Prandtl number, s, enhances the maximum growth rate for small and moderate values of x 0 for both modes, but the precise range of influence depends on the parity of the mode. For given values of x 0, s, sm, Qc, Bv, the plume can be destabilized by an increase in the inclination of the field to the vertical so that an inclined field may make the plume more unstable than a horizontal field of the same strength. To clarify the differing effects of magnetic field and diffusion on the two categories of modes, the preference of the four modes (vertical sinuous mode, Sv, the oblique sinuous mode, So, and the two corresponding varicose modes, Vv, Vo) has been summarized in regime diagrams in the x 0Ks plane for different values of the parameters sm, Qc, Bv. It is found that for any prescribed set of parameters sm, Qc, Bv, the growth rate as a function of the plume thickness 2x 0 and s attains its maximum along a curve with x 0 always occurring in the range 1.3!x 0!1.7. This overall maximum is sinuous for small s and varicose for large s. As compared with the stability of the single interface studied in part I, it is found that the finite thickness of the buoyant fluid has the effect of enhancing the instability. This had already been shown to be the case in the absence of the field (Eltayeb & Loper 1994), and motivated an investigation of the influence of the field on the instability of the non-magnetic Cartesian plume. It has been shown that for certain values of the parameters x 0, s, sm, Qc, Bv, the magnetic field has a destabilizing influence. Detailed investigations of the growth rate revealed that the enhancement of instability is caused by viscous diffusion. We recall that viscous diffusion enhances instability in the case of the non-magnetic plume, and here we find that the magnetic field, whether horizontal or inclined to the vertical, can enhance further the destabilizing influence of viscosity. This is found to be the result of the transfer of energy from the basic state horizontal temperature variations to the wave. Such destabilizing influence affects both varicose and sinuous modes. However, the strength of its effect on either mode depends on the values of the parameters x 0, s, sm, Qc, Bv. We wish to thank the referees for their constructive comments which led to the improvement of the original version of the paper.

Appendix A. Derivation of the expression for the growth rate of the magnetic Cartesian plume The coefficients of Re1, which result from substitution of equations (2.27) and (2.28) into equations (2.6), (2.8), (2.9) and (2.12) give the set


0; Dp1 K T1 Z 2inDwu

(A 1)

~ 1 T0 K inDTu 0 ; DT1 K w1 Z s½U

(A 2)

2630


~ 1 w0 K inDwu 0; Dw1 C idQb1 C T1 C np1 Z U

(A 3)

~ 1 b0 C inDwb x 0 ; Db1 C idw1 Z sm ½U

(A 4)

~ 1u 0; Du1 C idQbx1 K Dp1 Z U

(A 5)

~ 1 bx0 Dbx1 C idu1 Z sm U

(A 6)

~ is defined by equation (2.17) when UZU1. These equations are to be and U solved in the two distinct regions 0%x 0!x and x 0!x!N subject to the parity conditions (2.25) or (2.26) and the decay conditions (2.24). (i) The continuity conditions to be satisfied at xZx 0 consist of (2.24) (ii) while (iii) and (iv) are replaced by ) Dw1 and Dp1 continuous at x Z x0 : (A 7) U2 Z Kinu1 ðx0 Þ First consider (A 5) and (A 6) and define UG Z u1 GQ1=2 bx1 ;

(A 8)

so that the two equations can be combined into ~ 1 ½u 0 Gsm Q1=2 bx 0 : DUGGidQ1=2 UG Z Dp1 C U

(A 9)

This pair of equations is to be satisfied subject to the symmetry condition UGv ð0Þ Z 0 or DUGs ð0Þ Z 0;

(A 10)

UG and DUG are continuous at x Z x0

(A 11)

the continuity conditions

and decay conditions UGðxÞ/ 0

as x /N:

(A 12)

We wish to express U2 in terms of known functions. We will find it useful to note that 1 U2 Z K n½UCðx 0 Þ C UKðx 0 Þ: 2 The first step begins with the function " ( g ðxKx Þ )# eG 0 1 KgGðxCx0 Þ GG Z ; KP e C 4gG eKgGðxKx0 Þ Proc. R. Soc. A (2005)

gG Z

(A 13)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 C n 2 HidQ1=2 ; (A 14)

2631


where the upper (lower) expression applies for 0%x!x 0 (xOx 0) and P is defined by equation (1.4). It is straightforward to verify that this function satisfies 1 DUGGidQ1=2 UG Z K dc ðx K x0 Þ; (A 15) 2 where dc(x) is the generalized delta-function, as well as symmetry conditions (A 10) and the decay conditions (A 12). The result of multiplying (A 15) by GG, integrating from 0 to N and then integrating by parts twice is ðN (A 16) UGðx0 Þ Z K2 GG½DUGGidQ1=2 UGdx: 0

Substituting this into equation (A 13), using equation (A 9) and integrating by parts once, yields ðN U2 Z in I0 K p1 ðDGC C DGKÞdx ; (A 17) 0

where

ðN I0 Z

0

~ 1 fu 0 ½GC C GK C sm Q1=2 bx0 ½GC K GKgdx: U

(A 18)

Note that the only unknown factor in equation (A 17) is p1. The next (second) step is to express the integral involving p1 in terms of known functions. A single equation for p1 may be obtained by forming ðD2 C d2 QC 1ÞDðA 1ÞC 2 ðD C d2 QÞðA 2ÞC DðA 3ÞK idQðA 4Þ; Lp1 Z Ftot ;

(A 19)

in which 9 > > =

L hD4 C ð1 C d2 QÞD2 C n 2 D; ~ 1 w0 K inDwu 0 Þ C DðU 0Þ Ftot Z in½D2 C d2 Q C 1DðDwu ~ 1 T0 K inDTu 0 Þ K idQsm ðU ~ 1 b0 C inDwb x0 Þ: CsðD2 C d2 QÞðU Now let us consider the function 2

SH Z n ½DGC K DGK C idQ

1=2

" DGC C DGK K

3 X

> > ;

(A 20)

# Cj Hj ðxÞ ;

(A 21)

jZ1

in which " ( l ðxKx Þ )# j 0 e 1 ; P eKlj ðxCx0 Þ C Hj Z 2 KeKlj ðxKx0 Þ Cj Z

d2 Qmj C n 2 ½1 C m2j ; 3n 2 C 2½1 C d2 Qmj

(A 22)

(A 23)

where lj is given by equation (2.32). It is tedious but straightforward to verify that LSH Z KidQ1=2 ðd2 Q C n 4 Þ½DGC C DGK Proc. R. Soc. A (2005)

(A 24)

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and that, due to the structure of Cj, the functions SH, DSH, DSH, DDSH, D2SH, DD2SH and D3SH are all continuous across xZx 0. Substituting equation (A 24) into equation (A 17), integrating from xZ0 to N, integrating by parts and using equation (A 19), we have ðN n SH Ftot dx: (A 25) U2 Z inI0 C 1=2 4 Q ðn C d2 QÞ 0 The use of equations (A 18) and (A 20) leads to the expression (2.35), where ðN ~ 1 u0 ½GC CGKdx U20 Zin U 0 ðN in ~ 1 w0 CinD wu 0 Þdx ½n 2 ðDGC CDGKÞCidQ1=2 ðDGC KDGKÞðU K 4 2 n Cd Q 0 ð 3 N X in ~ 1 w0 Cinð1C2d2 QC2m2j ÞðDwu 0 Þgdx: C mj Hj fU K 4 j n Cd2 Q jZ1 0 (A 26) ðN 3 X in ~ 1 T0 K inDTu 0 Þdx Usi Z K 2 Cj Hj ½m2j C d2 QðU 4 ½d Q C n jZ1 0 and

ðN

Usm Zin K

0

(A 27)

~ 1 bx0 ½GC KGK dx U

inQ1=2 n4 Cd2 Q

ðN 0

~ 1 b0 CinDwb x0 Þdx ½idQ1=2 ðDGC CDGK ÞCn2 ðDGC KDGK ÞðU

ðN 3 ndQ X ~ 1 b0 CinDwb x0 Þdx; K 4 Cj Hj ðU n Cd2 Q jZ1 0

ðA 28Þ

in which GG is given by equation (A 14) and the zeroth-order perturbations by equations (2.29)–(2.32). The expression for U2 reduces to that obtained for the non-magnetic case when Qc/0; it reduces to the magnetic single interface expression derived in part I when x 0/N, and U2 Z Oðx02 Þ when x 0/0.

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